Physics

Law of total covariance

Mathematics

Matrix norm

Covariance operator

Statistics

Singular value

Mathematical optimization

Matérn covariance function

Rational quadratic covariance function

Covariance function

Estimator

Eigenvalues and eigenvectors

Multivariate statistics

Covariance

Estimation of covariance matrices

Covariance matrix

CMA-ES

Quantum mechanics

Applied mathematics

Wishart distribution

Covariance intersection

The Annals of Statistics

We show that in a common high-dimensional covariance model, the choice of loss function has a profound effect on optimal estimation. In an asymptotic framework based on the Spiked Covariance model and use of orthogonally invariant estimators, we show that optimal estimation of the population covariance matrix boils down to design of an optimal shrinker η that acts elementwise on the sample eigenvalues. Indeed, to each loss function there corresponds a unique admissible eigenvalue shrinker η* dominating all other shrinkers. The shape of the optimal shrinker is determined by the choice of loss function and, crucially, by inconsistency of both eigenvalues and eigenvectors of the sample covariance matrix. Details of these phenomena and closed form formulas for the optimal eigenvalue shrinkers are worked out for a menagerie of 26 loss functions for covariance estimation found in the literature, including the Stein, Entropy, Divergence, Fréchet, Bhattacharya/Matusita, Frobenius Norm, Operator Norm, Nuclear Norm and Condition Number losses.

Optimal shrinkage of eigenvalues in the spiked covariance model